In a document, answer the following questions:

1. (a) Given operation is binary operation. As g1 ◦ g2 ∈ G and h1 ∗h2 ∈ H.
Hence (g1 ◦ g2, h1 ∗ h2) ∈ G×H.
(b) We have
(g1, h1).((g2, h2).(g3, h3)) = (g1, h1)(g2 ◦ g3, h2 ∗ h3)
= (g1 ◦ (g2 ◦ g3), h1 ∗ (h2 ∗ h3))
As G and H are group we have associative law on G and H. Hence
we have :
(g1 ◦ (g2 ◦ g3), h1 ∗ (h2 ∗ h3)) = ((g1 ◦ g2) ◦ g3, (h1 ∗ h2) ∗ h3)
= ((g1, h1).(g2, h2)).(g3, h3)
Hence operation is associative in G×H.
(c) If eG and eH are identity of G and H respectively, we have (eG, eH)
is identity of G×H.
(d) If inverse of g is denoted by g−1 and inverse of h is denoted by h−1,
then inverse of (g, h) is given by (g−1, h−1). Hence inverse exists.
Hence G×H is group.
2.
Z2 × Z2 × Z2
In all subgroups operation is component wise addition modulo 2.
(a) Z2 × {0} × {0}
(b) {0} × Z2 × {0}
(c) {0} × {0} × Z2
(d) {(0, 0, 0), (1, 1, 1)}
(e) {(0, 0, 0), (1, 1, 0)}
(f) {(0, 0, 0), (0, 1, 1)}
(g) {(0, 0, 0), (1, 0,...